Advertisements
Advertisements
Question
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Solution
Let I = `int_0^(pi/4) (dx)/(1 + tanx)`
= `int_0^(pi/4) (dx)/(1 + sinx/cosx)`
= `int_0^(pi/4) (cos x dx)/(cosx + sinx)`
= `1/2 int_0^(pi/4) (2cosx)/(cosx + sinx) dx`
= `1/2 int_0^(pi/4) (cosx + sinx + cosx - sinx)/(cosx + sinx) dx`
= `1/2 [int_0^(pi/4) (cosx + sinx)/(cosx + sinx) dx + int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx]`
= `1/2 [int_0^(pi/4) 1dx + int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx]`
= `1/2 (I_1 + I_2)`
Where, I1 = `int_0^(pi/4) 1dx`
= `[x]_0^(pi/4) = pi/4`
And I2 = `int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx`
Let cosx + sinx = t
⇒ (–sinx + cosx)dx = dt
When x = 0, t = 1
And x = `pi/4`, t = `2/sqrt(2)`
∴ I2 = `int_1^(2/sqrt(2)) (dt)/t`
= `[logt]_1^(2/sqrt(2))`
= `log 2/sqrt(2) - log 1`
= `log 2/sqrt(2) - 0`
= `log2^(3/2)`
= `3/2 log 2`
∴ I = `1/2(I_1 + I_2)`
or I = `1/2(pi/4 + 3/2 log 2)`
APPEARS IN
RELATED QUESTIONS
Prove that:
`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+c`
Integrate the function in x sin 3x.
Integrate the function in `x^2e^x`.
Integrate the function in x sin-1 x.
Integrate the function in tan-1 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x tan^-1 x .dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
`int"e"^(4x - 3) "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int 1/sqrt(x^2 - 9) dx` = ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Solve: `int sqrt(4x^2 + 5)dx`
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
